The first part, General Introductory Remarks,
distinguishes between various approaches to algebra, namely
`Practical', `Philological' and `Theoretical'. Hamilton then
gives his reasons for contending that Algebra may be regarded as
the Science of Pure Time.

The second part, On Algebra as the Science of Pure Time
conceives of real numbers of ratios of steps between moments
of time, and derives the basic properties of the algebra of real
numbers from this conception. The existence of square roots of
positive numbers, of $n$th roots, of exponentials, and of
logarithms is discussed.

The third part, The Theory of Conjugate Functions, or
Algebraic Couples, defines complex numbers as `algebraic
couples', which are ordered pairs of real numbers, with
appropriately defined operations of addition, subtraction,
multiplication and division. Hamilton gives a careful proof of
basic properties of the exponential function, and discusses the
nature of logarithms, providing a natural framework to justify
results obtained by his friend John T. Graves concerning complex
logarithms which had been questioned by eminent British
mathematicians such as Peacock and Herschel.